Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 218, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
CRITICAL POINTS AND CURVATURE IN CIRCULAR
CLAMPED PLATES
JAIME ARANGO, ADRIANA GÓMEZ, ANDRÉS SALAZAR
Abstract. In this article we investigate some qualitative properties of the
solutions of the classical linear model for clamped plates on circular domains,
under constant sign external loads. In particular we prove that inside the circle
there are at most a finite number of critical points, which in turn rules out the
existence of critical curves. We also study the curvature of the level curves of
the solutions, and we prove that the curvature function is continuous up to
the border, even though the gradient of the solutions vanishes on the border
circle.
1. Introduction
The classical model for the deflection u of a clamped plate under an external
load is given by
∆2 u = f in B,
(1.1)
∂u
u=
= 0 on ∂B,
∂ν
where ∆2 is the biharmonic operator, B is a planar domain, f is the density of the
external load and ∂u
∂ν is the outward normal derivative of u at the boundary ∂B.
It is well known that problem (1.1) possesses exactly one solution u ∈ C 4 (B) ∩
C 2 (B̄) provided f ∈ C(B̄), see for example [10]. Moreover, for some special domains
the Green function of problem (1.1) has been explicitly computed, as it is the case of
the disk ([5]) and the limaçon ([7]). However, in contrast with existence, uniqueness
and regularity, the geometric properties of the solutions seem to be not so well
documented. Unlike second order elliptic operators, the maximum principle does
not hold for the clamped plate problem, and as a consequence, the sign preserving
property (SPP) for a domain B associated to problem (1.1),
f ≥ 0 in B
implies
u ≥ 0 in B,
does not hold for general domains. The famous 1908 Boggio-Hadamard conjecture
claims that the SPP applies on convex domains. Boggio [5] proved in 1905 that the
SPP holds for circular domains. However, in 1949 Duffin [8] proved the conjecture
to be false for infinitely long rectangles and in 1951 a result due to Garabedian
[9] showed that the SPP does not hold for eccentric enough ellipses. Since then,
2000 Mathematics Subject Classification. 35J40, 74K20.
Key words and phrases. Clamped plates; critical points; curvature.
c
2014
Texas State University - San Marcos.
Submitted August 26, 2014. Published October 16, 2014.
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J. ARANGO, A. GÓMEZ, A. SALAZAR
EJDE-2014/218
many other counterexamples have been given in the literature. A strikingly simple,
explicit example, of a function u that changes sign and satisfies (1.1), with f > 0
and B a certain ellipse, was provided by Shapiro [16] in 1994.
The critical set K of a function u is the set of its critical points, that is to say,
the points with vanishing gradient. The description of K can be a key step in the
way to describe the geometric behavior of solutions u to (1.1). To the authors’
knowledge, there are few results regarding the structure of this set. In a 1994 paper
Soranzo [17] proved that if B is a disk and f is a radially symmetric, nonnegative,
nonzero function, then, inside of B, the only critical point of the solution u to (1.1)
is the center of B. We also refer to the results of Grunau and Sweers [11], who
proved that when B is a disk, the solution u to (1.1) possesses no minima in the
interior of B, provided that f is a nonzero, nonnegative function.
In a recent article two of the present authors [3], studied the critical set of solutions to equations that model the deflection of membranes fixed at the border,
subject to the action of a constant sign analytical external force (see also [1]). According to this work the critical set associated to the solutions of the corresponding
second order elliptic boundary-value problem, is made up of finitely many critical
points and finitely many Jordan critical curves. Although the authors think that
analogue results should hold for the biharmonic problem (1.1), at least for domains
where the SPP holds, a proof is not known to them. In this paper we follow techniques similar to the ones employed in [3], to show that when B is a disk, f is a
nonzero, nonnegative, real analytic function, and u is a solution to (1.1), the critical
set of u, inside of B, is made up of finitely many isolated critical points. According
to [11], critical points should be either maxima or saddle points.
We remark that the analyticity condition on f grants that the solution u to (1.1)
is analytic (see for example [15]), which in turn allows us to apply some general
results obtained in previous works (see [4], [3] and [2]), concerning the structure of
the critical set of certain analytical functions.
We also analyze the nodal sets of the directional derivatives of the solution to
(1.1), in order to show that, whenever the density of the external load f is real
analytic, the curvature function on the level curves of u can be extended to the
border of the disk, even though ∇u vanishes there. Further, we present an explicit
example showing that when f changes sign, it is possible that the curvature function
fails to be continuous on the border. We do not know of any previous analysis
regarding the curvature of level sets of solutions to problem (1.1).
2. Radial solutions and critical points to the clamped plate
equation
Let us summarize some known results concerning the qualitative properties of
the solution to problem (1.1), under the assumption that f is a continuous, nonzero,
nonnegative function.
Theorem 2.1 (Boggio [5], 1904). Sign preserving property holds when B is a disk.
Theorem 2.2 (Grunau-Sweers [11], 2001). If f is a nonzero, nonnegative function,
the solution to (1.1) possesses no local minima when B is a disk.
In contrast with the above results, the following property holds for more general
domains.
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CRITICAL POINTS AND CURVATURE
3
Lemma 2.3. If the SPP holds for a domain B whose boundary satisfies the interior
ball condition, and f is a nonzero, nonnegative function, then the solution u to
problem (1.1) satisfies ∆u|∂B ≥ 0.
Proof. Let us assume that ∆u(q) < 0 at some point q ∈ ∂B. Let D be a disk inside
of B, tangent to ∂B at q and such that ∆u < 0 on D. Since the SPP holds, it follows
that u ≥ 0 on B. Hopf’s boundary point lemma then implies that ∂u
∂ν (q) < 0, which
∂u
contradicts the fact that ∂ν = 0 on ∂B.
However when B is a disk and u is a radial solution, the inequality in Lemma
2.3 can be proved to be strict. This and some other properties of radial solutions
are summarized in Lemma 2.4 and Corollary 2.5, which for the most part follow
[17, Proposition 1].
Lemma 2.4. Let B ≡ Bρ (0) be the disk of radius ρ centered at the origin and
let u ∈ C 4 (B) ∩ C 2 (B̄), be a radial nontrivial solution of (1.1), with f a nonzero,
nonnegative function. Then u is strictly positive in B, w(|x|) = u(x) satisfies
w0 < 0 on (0, ρ), ∆u(0) < 0, and ∆u|∂B > 0.
Proof. The first three statements correspond to Proposition 1 in [17]. To prove that
∆u|∂B > 0, notice that if u is a radially symmetric function, and w(|x|) = u(x),
0
then ∆u = 1r (rw0 )0 so that, if W = 1r (rw0 ) , it follows, according to equation (51)
0
in [17], that W (r) ≥ 0 on (0, ρ). Moreover, by Lemma 2.3, W (ρ) ≥ 0. However
W (ρ) = 0 would imply W (r) ≤ 0 on (0, ρ), and, as u = 0 on ∂B, the maximum
principle yields u > 0 on B. In that case, ∂u
∂ν < 0 on ∂B according to Hopf boundary
point Lemma, thus contradicting the boundary condition ∂u
∂ν = 0.
As an incidental consequence of Lemma 2.4 we can fully describe the critical
set of any radially symmetric solution to problem (1.1) on a disk, as stated in the
following corollary.
Corollary 2.5. If f is a nonzero, nonnegative function, the center of the disk is
the only interior critical point of a radially symmetric solution to (1.1).
To finish, we quote a result that characterizes the critical set of any semi-Morse
function. Following [3], we say that a critical point of a function v is semi–Morse,
if the Hessian matrix of v does not vanish at that point. A function is termed
semi–Morse if all of its critical points are semi-Morse. The structure of the critical
set of analytical semi-Morse functions was studied in [4] and [3].
Lemma 2.6 ([3, Lemma 2]). Let B ⊂ R2 be a planar domain with smooth boundary
∂B, and let v be a real analytic, semi–Morse function defined on an open neighborhood of B̄. If all of the critical points of v belong to B, then the critical set of v is
made up of finitely many isolated critical points, and finitely many regular analytic
Jordan curves.
In the next section we will show that solutions u to (1.1) are semi-Morse provided
B is a disk. Notice that this result heavily relies on the SPP exhibited by the
domain.
3. Isolated critical points
Given a solution u to problem (1.1) on the unit disk, we can follow Grunau and
Sweers (see [11]) and consider the Möebius transformations on the unit complex
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J. ARANGO, A. GÓMEZ, A. SALAZAR
EJDE-2014/218
disk B
a−x
,
1 − āx
where x ∈ B, a ∈ B, in order to define the function v as
1
v(x) = 0
u(ha (x)).
|ha (x)|
ha (x) =
(3.1)
We notice that
1 1 ∆(u ◦ ha )(x)
+
u(h
(x))
∆
+
2
∇
· ∇(u ◦ ha )(x).
a∆v(x) =
a
|h0a (x)|
|h0a (x)|
|h0a (x)|
On the other hand, taking into account that ha is a conformal map, we have that
∆(u ◦ ha )(x) = |h0a (x)|2 ∆u(ha (x)).
Therefore in the particular case that ha (x) turns to be a critical point of u, it follows
that
1 ∆v(x) = |h0a (x)| ∆u(ha (x)) + u(ha (x)) ∆
.
(3.2)
|h0a (x)|
Grunau and Sweers [11, Lemma 1] showed that v satisfies:
∆2 v = |h0a |3 (f ◦ ha )
in B,
(3.3)
∂v
= 0 on ∂B.
∂ν
Moreover, they also showed (see [11], equations (6) and (7)), that the function
Z
1
w(x) ≡
v(|x| z) ds(z), x ∈ B,
(3.4)
2π |z|=1
v=
solves the equation
∆2 w = g
w=
where
1
g(x) =
2π
∂w
=0
∂ν
Z
in B,
(3.5)
on ∂B,
|h0a |3 f ◦ ha (|x| z) ds(z).
|z|=1
The function w can be seen as the radial average of v. Since w is radial, we can
consider it as a single variable function, w(r) = w(x), for r = |x|.
Next we show that when B is a disk, interior critical points of solutions to (1.1)
turn to be semi-Morse.
Lemma 3.1. Let B be the unit disk, and let u be a solution to (1.1), with f a
nonnegative, nonzero function. If a ∈ B is a critical point of u then ∆u(a) < 0.
Proof. A straightforward computation shows that
Z
1
w00 (r) =
Hv (rz) z · z ds(z),
2π |z|=1
where v and w are defined by (3.1) and (3.4) respectively, and Hv stands for the
Hessian matrix of v. Therefore
Z
1
1
w00 (0) =
Hv (0) z · z ds(z) = ∆v(0).
2π |z|=1
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CRITICAL POINTS AND CURVATURE
5
Notice that, as w is a radial solution of the boundary-value problem (3.5), then,
according to Lemma 2.4, we have ∆w(0) < 0. Since ∆v(0) = 2 w00 (0), it follows
that ∆v(0) < 0.
Now, taking into account (3.2), and the fact ha (0) = a, it follows that in the
case a is a critical point of u,
∆v(0) = (1 − |a|2 )∆u(a) +
4 |a|2
u(a).
1 − |a|2
The conclusion follows from the above identity, given that, as B has the sign preservation property (Theorem 2.1), u(a) ≥ 0.
It is known that locally the critical set of an analytic, semi-Morse function is
either an isolated point or a curve (see [4], [3], [2]). The above lemma proves that
u is semi–Morse in the interior of B, thus yielding the local structure of the critical
set in the interior of B. We will prove that curves of critical points are precluded
and that critical points do not accumulate on the boundary of B.
Lemma 3.2. Let B be the unit disk and let a ∈ B. If u is a radial solution to (1.1)
(with f a nonnegative, nonzero function), and v(x) is given by (3.1), then ∆v > 0
on ∂B.
Proof. Notice that every point on ∂B is critical for u, and that ∂B is invariant
under ha . The result now follows from (3.2), given that u vanishes on ∂B, and, by
Lemma 2.4, ∆u(ha (x)) > 0.
Theorem 3.3. If B is a disk, solutions u to (1.1) have no critical curves, whenever
f is an analytic, nonnegative, nonzero function.
Proof. Let us assume that Γ is a curve of critical points of u. We know from Lemma
2.6 that Γ is smooth, so let τ and η be respectively tangent and normal unitary
vectors to Γ at some given point p. We can see that if r = r(t) parametrizes Γ,
u(r(t)) is constant, from which it follows that
Hu (r(t)) r0 (t) · r0 (t) + ∇u (r(t)) · r00 (t) = 0.
Moreover, as Γ is a curve of critical points, we conclude that Hu (p) τ · τ = 0, and
∆u(p) = Hu (p) η · η. From Lemma 3.1 it then follows that Γ is a curve of local
maxima, therefore there must be a minimum of u inside of Γ, thus contradicting
Theorem 2.2.
We remark that Theorem 3.3 does not hold if f changes sign. Let us define
u(x1 , x2 ) = (1 − x21 − x22 )2 (1 + 8x21 + 8x22 )2 .
A direct calculation shows that u satisfies Problem 1.1 in the unitary disk with
f = ∆2 u = 192 −5 + 24x21 + 24x22 changing sign in B. However it can be seen
that the circle x21 + x22 = 41 is a critical curve of u inside of B.
Lemma 3.4. If B is a disk, u is a solution to (1.1), and f is nonnegative and
nonzero, then ∆u|∂B > 0.
Proof. According to Lemma 2.3 we already know that ∆u|∂B ≥ 0. Let us suppose
there exists a point q ∈ ∂B such that ∆u(q) = 0. Now, as f is a nonzero, nonnegative function, we might fix a point a ∈ B such that f (a) > 0, and for > 0,
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J. ARANGO, A. GÓMEZ, A. SALAZAR
EJDE-2014/218
f (a) < 1 given, define the radial function
(
f (a) − |x| if |x| < f (a),
f (x) =
0
if |x| ≥ f (a).
However, as ha maps neighborhoods of 0 into neighborhoods of a, we can choose small enough to guarantee that f ◦ ha vanishes outside a small neighborhood of a,
and
f ≥ |h0a |3 (f ◦ ha ) in B.
We consider now the solution w to the boundary-value problem
∆2 w = f
in B,
∂w
= 0 on ∂B,
∂ν
and set v(x) = |h10 | w(ha (x)). We notice that v satisfies (3.3), if we replace f by f .
a
It follows that z ≡ u − v satisfies
w=
∆2 z ≥ 0
in B,
∂z
= 0 on ∂B.
∂ν
Now, as ∆u(q) = 0 and according to Lemma 3.2, ∆v(q) > 0, we would have
∆z(q) < 0, thus yielding a contradiction, as by Lemma 2.3 ∆z ≥ 0 in ∂B.
z=
Alternatively, we could resort to the well known Boggio’s formula for the Green
function of problem (1.1) in the disk, to prove the above Lemma in a more direct
way, though we have opted for this indirect approach.
Lemmas 3.1 and 3.4 show that analytic solutions to problem (1.1) are semi-Morse
functions, hence the critical set should be as described by Lemma 2.6. Further, as
we state in the next theorem, the critical set do not include critical curves.
Theorem 3.5. If B is a disk, f is an analytic, nonnegative, nonzero function, and
u is a solution to (1.1), the set of critical points of u inside of B is made up of
finitely many points. Moreover critical points are either maxima or saddle points.
Proof. Lemmas 3.1 and 2.6, and Theorem 3.3, grants that critical points of a solution u to (1.1) are isolated. To prove that there are only finitely many critical
points in the interior of B, it only remains to prove that they do not accumulate on
the boundary. To see this, let us assume that (ak ) is a sequence of critical points in
B, such that ak → a, with a a point in the the boundary of B. However this would
imply that ∆u(ak ) → ∆u(a) and ∆u(a) ≤ 0, given that by Lemma 3.1 ∆u(ak ) < 0,
thus contradicting Lemma 3.4. The fact that critical points are either maxima or
saddle points is just a consequence of a Grunau and Sweers result (see Theorem
2.2).
Example 3.6. Let B denote the disk of radius 1 centered at the origin. Define
u1 (x1 , x2 ) =
19(1 − x21 − x22 )2
64(100 − 180x1 + 81x21 + 81x22 )
and set u2 (x1 , x2 ) = u1 (−x1 , x2 ). A straightforward calculation shows that u =
u1 + u2 satisfies ∆2 u ≥ 0 in B and u = ∂u
∂ν = 0 on ∂B. Moreover u possesses
exactly three critical points inside
B:
a
saddle
at (0, 0) and two maxima at (−p, 0)
√
and (p, 0), where p2 = 73/54 − 30761/162. Figure 1 pictures the critical points of
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CRITICAL POINTS AND CURVATURE
7
u, as well as its level curves. Notice that the curvature of the level curves behaves
nicely near the boundary. This turns to be the case for solutions of (1.1), whenever
f is nonzero, nonnegative, as we show in the next section. However this behavior
is not granted if f = ∆2 u does not have a constant sign, as we will see later (see
Example 4.5).
Figure 1. Critical set and level curves of the function u given in
Example 3.6. Notice the curvature of the level sets near the border
circle.
4. Curvature of the level sets
In this section we will show that the function giving the curvature of the level
curves associated to the solution u to (1.1), can be extended to the boundary of
the disk. This claim is non-obvious since at points of non-vanishing gradient, the
curvature is given by
κ(x) = −
Hu θ(x) · θ(x)
,
|∇u(x)|
θ(x) = J
∇u(x)
,
|∇u(x)|
(4.1)
with J the −π/2 rotation matrix. According to Theorem 3.5, if f is analytic the
above formula makes sense for all but finitely many points in B. However, as ∇u
vanishes on the border circle, it is not clear that this formula can be continuously
extended to that curve.
We say that f in problem 1.1 is real analytic in B̄ if there exists an analytic
extension of f to an open set including B̄. It is well known that in that case the
solution u to (1.1) can be analytically extended to an open domain including B̄
(see [15]).
The techniques in this section heavily rely on the description of the nodal sets
of the directional derivatives of the solution u to (1.1). Following [2], given a
direction θ, the derivative of u in the θ direction will be denoted uθ ; that is to say,
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J. ARANGO, A. GÓMEZ, A. SALAZAR
EJDE-2014/218
uθ (x) = ∇u(x) · θ. The nodal set of uθ is now defined to be the set Nθ where uθ
vanishes; i.e.,
Nθ = {x ∈ B̄ : uθ (x) = 0}.
Figure 2. Nodal lines Nθ for θ = 0 (red), π4 (green), π2 (blue) and
3π
4 (gray), associated to the function u in Example 3.6 (points in
the border circle belong to all Nθ ).
If for a point x ∈ Nθ , ∇uθ (x) 6= 0, the nodal set Nθ is locally a curve, and given
that ∇uθ (x) = Hu (x) θ, it follows that this curve satisfies the ODE
z 0 = JHu (z)θ,
(4.2)
with initial data z(0) = x. On the other hand when x is a critical point of uθ , the
local structure of Nθ can be quite involved. However, if x is a Morse point of uθ
(that is, when the Hessian matrix of uθ is non singular at x), the local structure
of Nθ can be easily obtained from elementary Morse Theory. In particular, if
det JHuθ (x) < 0, Nθ must be locally homeomorphic to the level set of the function
f (x, y) = xy at the origin. It follows that, locally, Nθ is the union of the stable and
unstable manifolds of (4.2) at the equilibrium x. Our task now is to compute the
eigenvalues and the eigenspaces of JHuθ (x), in order to approximate the
stable and
unstable manifolds of (4.2) at x. To start with, for a given v ∈ C 2 B̄ and θ ∈ S 1
we denote
Hv,θ (x) = D(Hv (x)θ),
(4.3)
2
D being the standard derivative of a R value function. For the reader’s convenience
we write the full expression of Hv,θ for x = (x1 , x2 ) and θ = (cos θ, sin θ) (notice
that we use of the same letter to denote the direction θ and its argument):
cos θ vx1 x1 x1 + sin θ vx1 x1 x2 cos θ vx1 x1 x2 + sin θ vx1 x2 x2
Hv,θ (x) =
.
cos θ vx1 x1 x2 + sin θ vx1 x2 x2 cos θ vx1 x2 x2 + sin θ vx2 x2 x2
The matrix Hv,θ (x) turns out to be symmetric and to satisfy the commuting property
Hv,θ (x)α = Hv,α (x)θ, α, θ ∈ S 1 .
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CRITICAL POINTS AND CURVATURE
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Now, given a solution u to (1.1), we write H(x) and Hθ (x) instead of Hu (x) and
Hu,θ (x). With this simplified notation we have
Huθ (x) = Hθ (x).
Lemma 4.1. Let f in Problem (1.1) be a nonzero, nonnegative function, that is real
analytic in B̄. Let γ(t) = (cos t, sin t), t ∈ [0, 2π), be the standard parametrization
of ∂B by arc length and set θ(t) = γ 0 (t). Then for all t ∈ [0, 2π), setting x = γ(t)
and θ = θ(t), we have
∇(det H(x)) = ∆u(x)Hθ (x)θ
(4.4)
Hθ (x) θ = ∆u(x)Jθ,
(4.5)
2
det Hθ (x) = −(∆u(x)) .
(4.6)
Proof. Recall that if α and β are orthogonal directions, and A is any 2×2 symmetric
matrix,
2
det A = (Aα · α) (Aβ · β) − (Aα · β) and Tr A = Aα · α + Aβ · β.
Therefore, if A = H(x), we can deduce that
∇(det H(x)) = (H(x)β · β)Hα (x)α + (H(x)α · α)Hβ (x)β − (2H(x)α · β)Hα (x)β.
Now, given that ∇u(γ(t)) = 0 for t ∈ [0, 2π), we have
H(γ(t))θ(t) = 0.
(4.7)
Therefore, if we set x = γ(t), α = θ(t) = θ and β = Jθ, we would have
∇(det H(x)) = (H(x)β · β)Hα (x)α.
Moreover, given that ∆u(x) = Tr H(x) = H(x)β · β, equation (4.4) follows.
We can now derive (4.7) to obtain (4.5), and multiplying this last one by Hθ (x)J,
we deduce that
det(Hθ (x))Jθ = −(∆u(x))2 Jθ,
and the final claim follows.
Taking into account (4.7), we notice that points in ∂B, satisfy det H(x) = 0. As
a consequence of Lemma 4.1 it follows that, locally, the the nodal set of det H(x)
coincides with ∂B. We had already noticed that Nθ is related to the stable and
unstable manifolds associated to (4.2). Next we are going to take advantage of the
well known geometric regularity of these manifolds (see for example [14]), to further
study the structure of the nodal set Nθ .
Lemma 4.2. Let f be a real analytic, nonzero, nonnegative function, defined on
B̄, let ∂B be counterclockwise oriented, and denote by H the Hessian matrix of the
solution u to problem (1.1). Then, if x is a point in ∂B and θ is the unitary tangent
vector to ∂B at x, it follows that x is a saddle equilibrium of
z 0 = JH(z)θ,
(4.8)
and ∂B coincides with the stable manifold of (4.8) at x. Moreover, there is an open
neighborhood V of ∂B, such that for every point z ∈ V ∩ B, there exists exactly one
point x ∈ ∂B, such that z belongs to the unstable manifold of (4.8) at x, and the
backward orbit of z stays in V .
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J. ARANGO, A. GÓMEZ, A. SALAZAR
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Proof. Let x be a point on ∂B, and notice that JHθ (x) = JHuθ (x) is the linearization of (4.8), at x.
According to (4.7) x is an equilibrium of (4.8), and, taking into account (4.6), x
turns to be a saddle. Moreover, by (4.5)
JHθ (x) θ = −∆u(x) θ,
hence, recalling Lemma 3.4, −∆u(x) is a negative eigenvalue, with θ as associated
eigenvector. Furthermore, we had already noticed that at every saddle of (4.8), the
union of the stable and the unstable manifolds equals Nθ . Given that ∂B ⊂ Nθ ,
the stable manifold at x must coincide with ∂B.
Now, (4.6) implies that ∆u(x) is the positive eigenvalue at x. On the other hand,
as Hθ (x)2 = Tr(Hθ (x))Hθ (x) − det Hθ (x) I and, by equation (4.5), Hθ (x)2 θ =
∆u(x)Hθ (x)Jθ, we readily obtain
JHθ (x) Jθ = − Tr (Hθ (x)) θ + ∆u(x)Jθ.
(4.9)
Thus, as we already know how the matrix JHθ (x) transforms the orthogonal basis
{θ, Jθ}, it is not difficult to find the eigenspace associated to the eigenvalue ∆u(x).
In fact a straightforward calculation shows that
JHθ (x)α = ∆u(x)α,
with α = − Tr (Hθ (x)) θ + 2∆u(x)Jθ.
(4.10)
Notice now that (4.8) defines a system of differential equations depending on the
parameter θ. We know that given a direction θ there exists a point xθ ∈ ∂B such
that θ is tangent to ∂B at xθ . Furthermore, xθ is a saddle of the system associated
to θ, having unstable manifold Uθ .
As it is known (see for instance [6]), the unstable manifold depends smoothly on
θ, so that, for fixed θ0 we might locally parametrize Uθ by a C ∞ function w(t, θ),
defined on an open neighborhood (−, ) × (θ0 − δ, θ0 + δ), satisfying w(0, θ) = xθ .
Moreover, w can be chosen so that ∂w
∂t (0, θ0 ) = αθ0 , where αθ0 is given by (4.10),
with θ = θ0 and x = xθ0 . As xθ = Jθ we can readily compute the Jacobian
determinant of w at (0, θ0 ) to find out it is equal to −2∆u(xθ0 ). It follows that
w defines a local diffeomorphism, onto an open neighborhood of xθ0 . Notice that,
according to Theorem 3.5, this neighborhood can be granted to be small enough
so that does not contain critical points of u. Next, we can choose finitely many of
these sets to cover ∂B, and let V be the union of these neighborhoods.
It should be clear now that for every point z ∈ V ∩ B there exists θ such that
z ∈ Uθ and the backward orbit of z under (4.8) stays in V . Moreover, z cannot
belong to Uθ0 if θ0 is noncollinear with θ, given that Uθ and Uθ0 only could meet at
interior critical points of the solution u. However, it is still possible that z belongs
to U−θ , but the associated backward orbit could not stay in V .
We had already noticed that ∇u does not define a normal direction on ∂B, given
that points on this curve are critical. The next lemma allows us to define a vector
function that coincides with the direction of ∇u in points near the border of the
circle, and with the inward normal vector at boundary points.
Lemma 4.3. Let f be a real analytic, nonzero, nonnegative function, defined on
B̄, let u be the solution to problem (1.1), and let θx denote the counterclockwise
oriented unitary tangent vector at x ∈ ∂B. There exists > 0 such that
(
∇u(x)
, 1 − ≤ |x| < 1,
J |∇u(x)|
θ(x) =
θx ,
|x| = 1,
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CRITICAL POINTS AND CURVATURE
11
is continuous on 1 − ≤ |x| ≤ 1.
Proof. Let x ∈ ∂B be fixed and let θx be the unitary tangent vector to ∂B at x.
Consider now a sequence (zn ) in B such that zn → x. According to Lemma 4.2,
for zn close enough to ∂B, there exists a point xn ∈ ∂B, such that zn ∈ Uθn , with
θn the counterclockwise oriented tangent vector at xn . However, as Uθn ⊂ Nθn , it
follows that J∇u(z) points in the θn direction at every point z ∈ Uθn . Thus, we
have
∇u(zn )
= θn ,
J
|∇u(zn )|
and given that θn → θ, the conclusion follows.
Now we turn to the question of the continuity of the curvature function near ∂B.
Theorem 4.4. If u solves problem (1.1), with f a real analytic, nonzero, nonnegative function defined on B̄, then for all x ∈ ∂B, the curvature κ(z) defined in (4.1)
satisfies
lim κ(z) = 1.
z→x,z∈B
Proof. Let x be a point on ∂B and let θ be the (counterclockwise oriented) tangent
vector to ∂B at x, α the unstable direction at x. According to the generalized
L’Hopital rule in [13]
lim −
z→x
H(z)θ · θ
Dα (H(z)θ · θ)
= lim −
,
z→x
|∇u(z)|
Dα (|∇u(z)|)
(4.11)
whenever the latter limit exists. However
Hθ (z)α · θ
Dα (H(z)θ · θ)
=
,
∇u(z)
Dα (|∇u(z)|)
H(z) |∇u(z)|
·α
and as by Lemma 4.3
∇u(z)
= −Jθ,
|∇u(z)|
it follows that the limit in (4.11) exists. Moreover, as a consequence of (4.7),
H(x)Jθ = ∆u(x)Jθ, so that, taking (4.5) into account, we conclude that this limit
equals 1.
lim
z→x
We note that without the hypothesis about f , Theorem 4.4 may fail to be true.
The next example shows that if f changes sign, it could happen that the curvature
be discontinuous on ∂B.
Example 4.5. Let
19
u(x1 , x2 ) = (1 − x21 − x22 )2 (x1 − 1)2 + (x1 − 1)x2 + x22
10
A straightforward calculation shows that u satisfies (1.1) with f given by
32
f (x1 , x2 ) =
−10 − 60x1 − 57x2 + 114x1 x2 + 90x21 + 90x22
5
It can also be shown that u ≥ 0 on B and that f changes sign in B. Some
of the level curves of u can be seen in Figure 3: it can be adverted that there are
points arbitrarily close to the boundary point (1, 0) where the curvature is negative,
notwithstanding the fact that on ∂B, the curvature equals 1.
12
J. ARANGO, A. GÓMEZ, A. SALAZAR
EJDE-2014/218
Figure 3. Level curves of function u in Example 4.5.
Conclusions. We have shown that on a disk, critical sets of analytic solutions to
problem (1.1) have the same structure as for the analogous membrane deflection
problem. The analyticity assumption is rather strong, however it seems difficult
to avoid as our proofs heavily depend on the structure of the critical set of semi–
Morse functions, that was developed in an analytical setting (see [4]). Certainly the
analyticity, or the lack of it, affects the structure of the critical set, at least in the
case of solutions of second order elliptic equations, as shown by a trivial example
like
(
1,
if x2 + y 2 ≤ 41 ,
p
3
u(x, y) =
1 − 2 x2 + y 2 − 1 , if 14 < x2 + y 2 ≤ 1.
In the above case ∆u ≥ 0 in B, u = 0 on ∂B, but the critical set is the whole
disk x2 + y 2 ≤ 41 , instead of being a discrete set of points. In light of Corollary 2.5
an analogue behavior for radially symmetric solutions for the clamped plate model
is not possible, and the authors are not aware of examples shedding light on the
peculiarities exhibited by nonanalytic solutions to the problem (1.1).
On the other hand, techniques in this paper are tailored for disks, since our work
is grounded in the use of Möbius transformation defined on a disk. Nevertheless,
for the membrane deflection problem it is known that on simply connected planar
domains the critical set reduces to finitely many interior critical points. We expect
that at least some of the conclusions of Theorems 3.5 and 4.4 can be generalized for
domains satisfying the SPP. In a domain where the SPP does not hold, it is clear
that the last statement of Theorem 3.5 is not true. Actually, even for a constant
external force in (1.1), solutions of this model can change sign depending on the
domain, as shown by Grunau and Sweers in a recent paper [12].
Acknowledgements. The authors gratefully acknowledge the anonymous referee
for the careful revision of the manuscript and helpful suggestions. The authors
thank Universidad del Valle for financial support. A. Salazar also thanks Universidad Javeriana-Cali.
EJDE-2014/218
CRITICAL POINTS AND CURVATURE
13
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Jaime Arango
Universidad del Valle, Cali, Colombia
E-mail address: jaime.arango@correounivalle.edu.co
Adriana Gómez
Universidad del Valle, Cali, Colombia
E-mail address: adriana.gomez@correounivalle.edu.co
Andrés Salazar
Universidad Javeriana-Cali, Cali, Colombia.
Universidad del Valle, Cali, Colombia
E-mail address: andresmsalazar@javerianacali.edu.co